Filename: 308-counter-galois-onion.txt
Title: Counter Galois Onion: A New Proposal for Forward-Secure Relay Cryptography
Authors: Jean Paul Degabriele, Alessandro Melloni, Martijn Stam
Created: 13 Sep 2019
Last-Modified: 13 Sep 2019
Status: Open



1. Background and Motivation

    In Proposal 202, Mathewson expressed the need to update Tor's Relay
    cryptography and protect against tagging attacks. Towards this goal he
    outlined two possible approaches for constructing an onion encryption
    scheme that should be able to withstand tagging attacks. Later, in
    Proposal 261, Mathewson proposed a concrete scheme based on the
    tweakable wide-block cipher AEZ. The security of Proposal 261 was
    analysed in [DS18]. An alternative scheme was suggested in Proposal 295
    which combines an instantiation of the PIV construction from [ST14] and
    a variant of the GCM-RUP construction from [ADL17]. In this document we
    propose yet another scheme, Counter Galois Onion (CGO)
    which improves over proposals 261 and 295 in a number of ways. CGO has
    a minimalistic design requiring only a block cipher in counter-mode and
    a universal hash function. To take advantage of Intel's AES-NI and
    PCLMULQDQ instructions we recommend using AES and POLYVAL [GLL18]. In
    terms of security, it protects against tagging attacks while
    simultaneously providing forward security with respect to end-to-end
    authenticity and confidentiality. Furthermore CGO performs better than
    proposal 295 in terms of efficiency and its support of "leaky pipes".


1.2 Design Overview

    CGO makes due with a universal hash function while simultaneously
    satisfying forward security. It employs two distinct types of
    encryption, a dynamic encryption scheme DEnc and a static encryption
    scheme SEnc. DEnc is used for end-to-end encryption (layer n) and SEnc
    is used for the intermediate layers (n-1 to 1). DEnc is a Forward-
    Secure Authenticated Encryption scheme for securing end-to-end
    communication and SEnc provides the non-malleability for protecting
    against tagging attacks. In order to provide forward security, the key
    material in DEnc is updated with every encryption whereas in SEnc the
    key material is static. To support leaky pipes, in the forward
    direction each OR first attempts a partial decryption using DEnc and
    if it fails it reverts to decrypting using SEnc. The rest of the
    document describes the scheme's operation in terms of the low-level
    primitives and we make no further mention of DEnc and SEnc. However,
    on an intuitive level it can be helpful to think of:

    a) the combinations of E(KSf_I, *) and PH(HSf_I, *) as well as
    E(KDf_I, *) and PH(HDf_I, *) as two instances of a tweakable block
    cipher,

    b) the operation E(Sf_I, <0>) | E(Sf_I, <1>) |  E(Sf_I, <2>) | ... as a
    PRG with seed Sf_I,

    c) and E(JSf_I, <IV>) | E(JSf_I, <IV+1>) | ... | E(JSf_I, <IV+31>) as
    counter-mode encryption with <IV> as the initial vector.


2. Preliminaries

2.1. Notation

   Symbol               Meaning
   ------               -------
   M                    Plaintext
   Sf_I                 PRG Seed, forward direction, layer I
   Sb_I                 PRG Seed, backward direction, layer I
   Cf_I                 Ciphertext, forward direction, layer I
   Cb_I                 Ciphertext, backward direction, layer I
   Tf_I                 Tag, forward direction, layer I
   LTf_I                Last Tag, forward direction, layer I
   Tb_I                 Tag, backward direction, layer I
   LTb_I                Last Tag, backward direction, layer I
   Nf_I                 Nonce, forward direction, layer I
   LNf_I                Last Nonce, forward direction, layer I
   Nb_I                 Nonce, backward direction, layer I
   LNb_I                Last Nonce, backward direction, layer I
   JSf_I                Static Block Cipher Key, forward direction, layer I
   JSb_I                Static Block Cipher Key, backward direction, layer I
   KSf_I                Static Block Cipher Key, forward direction, layer I
   KSb_I                Static Block Cipher Key, backward direction, layer I
   KDf_I                Dynamic Block Cipher Key, forward direction, layer I
   KDb_I                Dynamic Block Cipher Key, backward direction, layer I
   HSf_I                Static Poly-Hash Key, forward direction, layer I
   HSb_I                Static Poly-Hash Key, backward direction, layer I
   HDf_I                Dynamic Poly-Hash Key, forward direction, layer I
   HDb_I                Dynamic Poly-Hash Key, backward direction, layer I
   ^                    Bitwise XOR operator
   |                    Concatenation
   &&                   Logical AND  operator
   Z[a, b]               For a string Z, the substring from byte a to byte b
                        (indexing starts at 1)
   INT(X)               Translate string X into an unsigned integer

2.2. Security parameters

   POLY_HASH_LEN -- The length of the polynomial hash function's output,
   in bytes. For POLYVAL, POLY_HASH_LEN = 16.

   PAYLOAD_LEN -- The longest allowable cell payload, in bytes (509).

   HASH_KEY_LEN -- The key length used to digest messages in bytes.
   For POLYVAL, DIG_KEY_LEN = 16.

   BC_KEY_LEN -- The key length, in bytes, of the block cipher used. For
   AES we recommend ENC_KEY_LEN = 16.

   BC_BLOCK_LEN -- The block length, in bytes, of the block cipher used.
   For AES, BC_BLOCK_LEN = 16.

2.3. Primitives

   The polynomial hash function is POLYVAL with a HASH_KEY_LEN-byte key. We
   write this as PH(H, M) where H is the key and M the message to be hashed.

   We use AES with a BC_KEY_LEN-byte key. For AES encryption (resp.,
   decryption) we write E(K, X) (resp., D(K, X)) where K is a BC_KEY_LEN-byte
   key and X the block to be encrypted (resp., decrypted). For an integer
   j, we use <j> to denote the string of length BC_BLOCK_LEN representing
   that integer.

2.4 Key derivation and initialisation (replaces Section 5.2.2)

   For newer KDF needs, Tor uses the key derivation function HKDF from
   RFC5869, instantiated with SHA256.  (This is due to a construction
   from Krawczyk.)  The generated key material is:

     K = K_1 | K_2 | K_3 | ...

   Where H(x, t) is HMAC_SHA256 with value x and key t
   and K_1     = H(m_expand | INT8(1) , KEY_SEED )
   and K_(i+1) = H(K_i | m_expand | INT8(i+1) , KEY_SEED )
   and m_expand is an arbitrarily chosen value,
   and INT8(i) is an octet with the value "i".

   In RFC5869's vocabulary, this is HKDF-SHA256 with info == m_expand,
   salt == t_key, and IKM == secret_input.

2.4.1. Key derivation using the KDF

   When used in the ntor handshake, for each layer I, the key material is
   split into the following sequence of contiguous values:

   Length             Purpose                    Notation
   ------             -------                    --------
   BC_KEY_LEN         forward Seed               Sf_I
   BC_KEY_LEN         backward Seed              Sb_I

   if (I < n) in addition derive the following static keys:

   BC_KEY_LEN         forward BC Key             KSf_I
   BC_KEY_LEN         backward BC Key            KSb_I
   BC_KEY_LEN         forward CTR Key            JSf_I
   BC_KEY_LEN         backward CTR Key           JSb_I
   HASH_KEY_LEN       forward poly hash key      HSf_I
   HASH_KEY_LEN       backward poly hash key     HSb_I

   Excess bytes from K are discarded.

2.4.2. Initialisation from Seed

   For each layer I compute E(Sf_I, <0>) | E(Sf_I, <1>) |  E(Sf_I, <2>) | ...
   and parse the output as:

   Length             Purpose                    Notation
   ------             -------                    --------
   BC_BLOCK_LEN       forward Nonce              Nf_I
   BC_KEY_LEN         forward BC Key             KDf_I
   HASH_KEY_LEN       forward poly hash key      HDf_I
   BC_KEY_LEN         new forward Seed           Sf'_I

   Discard excess bytes, replace Sf_I with Sf'_I, and set LNf_n and LTf_I
   to the zero string.

   Similarly for the backward direction, compute E(Sb_I, <0>) | E(Sb_I, <1>)
  | E(Sb_I, <2>) | ... and parse the output as:

   Length             Purpose                    Notation
   ------             -------                    --------
   BC_BLOCK_LEN       backward Nonce             Nb_I
   BC_KEY_LEN         forward BC Key             KDb_I
   HASH_KEY_LEN       forward poly hash key      HDb_I
   BC_KEY_LEN         new backward Seed          Sb'_I

   Discard excess bytes, replace Sb_I with Sb'_I, and set LNb_n and LTb_I
   to the zero string.

   NOTE: For layers n-1 to 1 the values Nf_I, KDf_I, HDf_I, Sf_I and their
   backward counterparts are only required in order to support leaky
   pipes. If leaky pipes is not required these values can be safely
   omitted.


3. Routing relay cells

   Let n denote the number of nodes in the circuit. Then encryption layer n
   corresponds to the encryption between the OP and the exit/destination
   node.


3.1. Forward Direction

   The forward direction is the direction that CREATE/CREATE2 cells
   are sent.


3.1.1. Routing From the Origin

   When an OP sends a relay cell, the cell is produced as follows:

   The OP computes E(Sf_n, <0>) | E(Sf_n, <1>) |  E(Sf_n, <2>) | ...
   and parses the output as

   Length             Purpose                    Notation
   ------             -------                    --------
   509                encryption pad             Z
   BC_BLOCK_LEN       backward Nonce             Nf'_I
   BC_KEY_LEN         forward BC Key             KDf'_I
   HASH_KEY_LEN       forward poly hash key      HDf'_I
   BC_KEY_LEN         new forward Seed           Sf'_I

   Excess bytes are discarded. It then computes the n'th layer ciphertext
   (Tf_n, Cf_n) as follows:

   Cf_n = M ^ Z
   X_n = PH(HDf_n, (LNf_n | Cf_n))
   Y_n = Nf_n ^ X_n
   Tf_n = E(KDf_n, Y_n) ^ X_n

   and updates its state by overwriting the old variables with the new
   ones.

   LNf_n = Nf_n
   Nf_n = Nf'_n
   KDf_n = KDf'_n
   HDf_n = HDf'_n
   Sf_n = Sf'_n

   It then applies the remaining n-1 layers of encryption to (Tf_n, Cf_n)
   as follows:

   For I = n-1 to 1:
     IV = INT(Tf_{I+1})
     Z  = E(JSf_I, <IV>) | E(JSf_I, <IV+1>) | ... | E(JSf_I, <IV+31>)
     % BC_BLOCK_LEN = 16
     Cf_I = Cf_{I+1} ^ Z[1, 509]
     X_I = PH(HSf_n, (LTf_{I+1} | Cf_I))
     Y_I = Tf_{I+1} ^ X_I
     Tf_I = E(KSf_I, Y_I) ^ X_I
     LTf_{I+1} = Tf_{I+1}

   Upon completion the OP sends (Tf_1, Cf_1) to node 1.


3.1.2. Relaying Forward at Onion Routers

   When a forward relay cell (Tf_I, Cf_I) is received by OR I, it decrypts
   it performs the following set of steps:

   'Forward' relay cell:

    X_I = PH(HDf_n, (LNf_I | Cf_I))
    Y_I = Tf_I ^ X_I
    if (Nf_I == D(KDf_I, Y_I) ^ X_I)  % cell recognized and authenticated
      compute E(Sf_I, <0>) | E(Sf_I, <1>) |  E(Sf_I, <2>) | ... and parse the
      output as Z, Nf'_I, KDf'_I, HDf'_I, Sf'_I

      M = Cf_n ^ Z
      LNf_I = Nf_I
      Nf_I = Nf'_I
      KDf_I = KDf'_I
      HDf_I = HDf'_I
      Sf_I = Sf'_I

      return M

    else if (I == n)    % last node, decryption has failed
      send DESTROY cell to tear down the circuit

    else    % decrypt and forward cell
      X_I = PH(HSf_I, (LTf_{I+1} | Cf_I))
      Y_I = Tf_I ^ X_I
      Tf_{I+1} = D(KSf_I, Y_I) ^ X_I
      IV = INT(Tf_{I+1})
      Z  = E(JSf_I, <IV>) | E(JSf_I, <IV+1>) | ... | E(JSf_I, <IV+31>)
      % BC_BLOCK_LEN = 16
      Cf_{I+1} = Cf_I ^ Z[1, 509]

      forward (Tf_{I+1}, Cf_{I+1}) to OR I+1

3.2. Backward Direction

   The backward direction is the opposite direction from
   CREATE/CREATE2 cells.

3.2.1. Routing From the Exit Node

   At OR n encryption proceeds as follows:

   It computes E(Sb_n, <0>) | E(Sb_n, <1>) |  E(Sb_n, <2>) | ...
   and parses the output as

   Length             Purpose                    Notation
   ------             -------                    --------
   509                encryption pad             Z
   BC_BLOCK_LEN       backward Nonce             Nb'_I
   BC_KEY_LEN         forward BC Key             KDb'_I
   HASH_KEY_LEN       forward poly hash key      HDb'_I
   BC_KEY_LEN         new forward Seed           Sb'_I

   Excess bytes are discarded. It then computes the ciphertext
   (Tf_n, Cf_n) as follows:

   Cb_n = M ^ Z
   X_n = PH(HDb_n, (LNb_n | Cb_n))
   Y_n = Nb_n ^ X_n
   Tb_n = E(KDb_n, Y_n) ^ X_n)

   and updates its state by overwriting the old variables with the new
   ones.

   LNb_n = Nb_n
   Nb_n = Nb'_n
   KDb_n = KDb'_n
   HDb_n = HDb'_n
   Sb_n = Sb'_n


3.2.2. Relaying Backward at the Onion Routers

   At OR I (for I < n) when a ciphertext (Tb_I, Cb_I) in the backward
   direction is received it is processed as follows:

   X_I = PH(HSb_n, (LTb_{I-1} | Cb_I))
   Y_I = Tb_I ^ X_I
   Tb_{I-1} = D(KSb_I, Y_I) ^ X_I
   IV = INT(Tb_{I-1})
   Z  = E(JSb_I, <IV>) | E(JSb_I, <IV+1>) | ... | E(JSb_I, <IV+31>)
   % BC_BLOCK_LEN = 16
   Cb_{I-1} = Cb_I ^ Z[1, 509]

   The ciphertext (Tb_I, Cb_I) is then passed along the circuit towards
   the OP.


3.2.2. Routing to the Origin

   When a ciphertext (Tb_1, Cb_1) arrives at an OP, the OP decrypts it in
   two stages. It first reverses the layers from 1 to n-1 as follows:

   For I = 1 to n-1:
     X_I = PH(HSb_I, (LTb_{I+1} | Cb_I))
     Y_I = Tb_I ^ X_I
     Tb_{I+1} = E(KSb_I, Y_I) ^ X_I
     IV = INT(Tb_{I+1})
     Z  = E(JSb_I, <IV>) | E(JSb_I, <IV+1>) | ... | E(JSb_I, <IV+31>)
     % BC_BLOCK_LEN = 16
     Cb_{I+1} = Cb_I ^ Z[1, 509]

   Upon completion the n'th layer of encryption is removed as follows:

   X_n = PH(HDb_n, (LNb_n | Cb_n))
   Y_n = Tb_n ^ X_n
   if (Nb_n = D(KDb_n, Y_n) ^ X_n)     % authentication is successful
     compute E(Sb_n, <0>) | E(Sb_n, <1>) |  E(Sb_n, <2>) | and parse the
     output as Z, Nb'_n, KDb'_n, HDb'_n, Sb'_n

     M = Cb_n ^ Z
     LNb_n = Nb_n
     Nb_n = Nb'_n
     KDb_n = KDb'_n
     HDb_n = HDb'_n
     Sb_n = Sb'_n

     return M

   else
     send DESTROY cell to tear down the circuit


4. Application connections and stream management

4.1. Amendments to the Relay Cell Format

   Within a circuit, the OP and the end node use the contents of
   RELAY packets to tunnel end-to-end commands and TCP connections
   ("Streams") across circuits. End-to-end commands can be initiated
   by either edge; streams are initiated by the OP.

   The payload of each unencrypted RELAY cell consists of:

       Relay command           [1 byte]
       StreamID                [2 bytes]
       Length                  [2 bytes]
       Data                    [PAYLOAD_LEN-21 bytes]

   The old Digest field is removed since sufficient information for
   authentication is now included in the nonce part of the payload.

   The old 'Recognized' field is removed. Instead a cell is recognized
   via a partial decryption using the node's dynamic keys - namely the
   following steps (already included in Section 3):

   Forward direction:

   X_I = PH(HDf_n, (LNf_I | Cf_I))
   Y_I = Tf_I ^ X_I
   if (Nf_I == D(KDf_I, Y_I) ^ X_I)  % cell is recognized and authenticated

   Backward direction (executed by the OP):

   If the OP is aware of the number of layers present in the cell there
   is no need to attempt to recognize the cell. Otherwise the OP can, for
   each layer, first attempt a partial decryption using the dynamic keys
   for that layer as follows:

   X_I = PH(HDb_I, (LNb_I | Cb_I))
   Y_I = Tb_I ^ X_I
   if (Nb_I = D(KDb_I, Y_I) ^ X_I)    % cell is recognized and authenticated

   The 'Length' field of a relay cell contains the number of bytes
   in the relay payload which contain real payload data. The
   remainder of the payload is padding bytes.

4.2. Appending the encrypted nonce and dealing with version-homogenic
     and version-heterogenic circuits

   When a cell is prepared to be routed from the origin (see Section
   3.1.1) the encrypted nonce N is appended to the encrypted cell
   (occupying the last 16 bytes of the cell). If the cell is prepared to
   be sent to a node supporting the new protocol, S is combined with other
   sources to generate the layer's nonce. Otherwise, if the node only
   supports the old protocol, n is still appended to the encrypted cell
   (so that following nodes can still recover their nonce), but a
   synchronized nonce (as per the old protocol) is used in CTR-mode.

   When a cell is sent along the circuit in the 'backward' direction,
   nodes supporting the new protocol always assume that the last 16 bytes
   of the input are the nonce used by the previous node, which they
   process as per Section 3.2.1. If the previous node also supports the
   new protocol, these cells are indeed the nonce. If the previous node
   only supports the old protocol, these bytes are either encrypted
   padding bytes or encrypted data.

5. Security and Design Rationale

   We are currently working on a security proof to better substantiate our
   security claims. Below is a short informal summary on the security of
   CGO and its design rationale.

5.1. Resistance to crypto-tagging attacks

   Protection against crypto-tagging attacks is provided by layers n-1 to
   1. This part of the scheme is based on the paradigm from [ADL17] which
   has the property that if any single bit of the OR's input is changed
   then all of the OR's output will be randomised. Specifically, if
   (Tf_I, Cf_I) is travelling in the forward direction and is processed by
   an honest node I, a single bit flip to either Tf_I or Cf_I will result
   in both Tf_{I+1} and Cf_{I+1} being completely randomised. In addition,
   the processing of (Tf_I, Cf_I) includes LTf_{I+1} so that any
   modification to (Tf_I, Cf_I) at time j will in turn randomise the value
   (Tf_{I+1}, Cf_{I+1}) at any time >= j . Thus once a circuit is tampered
   with it is not possible to recover from it at a later stage. This helps
   to protect against the standard crypto-tagging attack and variations
   thereof (Section 5.2 in [DS18]). A similar argument holds in the
   backward direction.


5.2. End-to-end authenticated encryption

   Layer n provides end-to-end authenticated encryption. Similar to the
   old protocol, this proposal only offers end-to-end authentication
   rather than per-hop authentication. However, CGO provides 128-bit
   authentication as opposed to the 32-bit authentication provided by the
   old protocol. A main observation underpinning the design of CGO is
   that the n'th layer does not need to be secure against the release of
   unverified plaintext (RUP). RUP security is only needed to protect
   against tagging attacks and the n'th layer does not help in that regard
   (but the layers below do). Consequently we employ a different scheme at
   the n'th layer which is designed to provide forward-secure
   authenticated encryption.


5.3 Forward Security

   As mentioned in the previous section CGO provides end-to-end
   authenticated encryption that is also forward secure. Our notion of
   forward security follows the definitions of Bellare and Yee [BY03] for
   both confidentiality and authenticity. Forward-secure confidentiality
   says that upon corrupting either the sender (or the receiver), the
   secrecy of the messages that have already been sent (or received) is
   still guaranteed. As for forward-secure authentication, upon corrupting
   the sender the authenticity of previously authenticated messages is
   still guaranteed (even if they have not yet been received). In order to
   achieve forward-secure authenticated encryption, CGO updates the key
   material of the n'th layer encryption with every cell that is
   processed. In order to support leaky pipes the lower layers also need
   to maintain a set of dynamic keys that are used to recognize cells that
   are intended for them. This key material is only used for partial
   processing, i.e. recognizing the cell, and is only updated if
   verification is successful. If the cell is not recognized, the node
   reverts to processing the cell with the static key material. If support
   for leaky-pipes is not required this extra processing can be omitted.


6. Efficiency Considerations

   Although we have not carried out any experiments to verify this, we
   expect CGO to perform relatively well in terms of efficiency. Firstly,
   it manages to achieve forward security with just a universal hash as
   opposed to other proposals which suggested the use of SHA2 or SHA3. In
   this respect we recommend using POLYVAL [GLL18], a variant of GHASH
   that is more compatible with Intel's PCMULQDQ instruction. Furthermore
   CGO admits a certain degree of parallelisability. Supporting leaky
   pipes requires an OR to first verify the cell using the the dynamic key
   material and if the cell is unrecognised it goes on to process the cell
   with the static key material. The important thing to note (see for
   instance Section 3.1.2) is that the initial processing of the cell
   using the static key material is almost identical to the verification
   using the dynamic key material, and the two computations are
   independent of each other. As such, although in Section 3 these were
   described as being evaluated sequentially, they can in fact be computed
   in parallel. In particular the two polynomial hashes could be computed
   in parallel by using the new vectorised VPCMULQDQ instruction.

   We are currently looking into further optimisations of the scheme as
   presented here. One such optimisation is the possibility of removing
   KDf_I and KDb_I while retaining forward security. This would further
   improve the efficiency of the scheme by reducing the amount of dynamic
   key material that needs to be updated with every cell that is processed.


References

[ADL17] Tomer Ashur, Orr Dunkelman, Atul Luykx, "Boosting Authenticated
Encryption Robustness with Minimal Modifications", CRYPTO 2017.

[BY03] Mihir Bellare, Bennett Yee, "Forward-Security in Private-Key
Cryptography", CT-RSA 2003.

[DS18] Jean Paul Degabriele, Martijn Stam, "Untagging Tor: A Formal
Treatment of Onion Encryption", EUROCRYPT 2018.

[GLL18] Shay Gueron, Adam Langley, Yehuda Lindell, "AES-GCM-SIV: Nonce
Misuse-Resistant Authenticated Encryption", RFC 8452, April 2019.

[ST13] Thomas Shrimpton, R. Seth Terashima, "A Modular Framework for
Building Variable-Input Length Tweakable Ciphers", ASIACRYPT 2013.
